Continuing in this direction, there is a close connection between coherence of an upper probability and the set of probability measures it dominates. In particular, [1]} proved a version of the celebrated upper envelope theorem, which states that a conditional upper probability is (W-)coherent if and only if it equals the upper envelope (or assertion-wise supremum) of some collection of ordinary conditional probabilities. It turns out that the notion of coherence in [2]} is generally stronger than W-coherence—“W” for Williams—so Walley's envelope theorem (Sec. 7.1.6) has only the “if” part of Williams's; see [2]} for details. This distinction is not important for us here because the set of conditional probability distributions is a critical component of our context. Indeed, the notion of dominance implied by (REF ) is relative to a particular conditional probability measure, so being an upper envelope of some collection of ordinary conditional probabilities is not enough to forge a connection between validity and coherence. Instead, we have to be explicit about the collection of probabilities. Let \({Q}\)  denote a user-specified set of candidate joint distributions \(\mathsf {Q}\)  for the process \(Y_1,Y_2,\ldots \) . Given \(Y^n=y^n\) , the upper envelope (REF ) of the conditional probabilities corresponding to members of \({Q}\)  is a coherent upper probability in the sense of Walley and, moreover, it satisfies strong validity.
